Question: Simplify. Rewrite the expression in the form $5^n$. $\dfrac{5^{10}}{5^3}=$
$\begin{aligned} \dfrac{5^{10}}{5^3}&=5^{10-3} \\\\ &=5^7 \end{aligned}$ This follows from the general rule $\dfrac{x^m}{x^n}=x^{m-n}$. Note that the powers have the same base. We can also see this is correct by expanding the powers. $\begin{aligned} \dfrac{5^{10}}{5^3}&=\dfrac{\overbrace{\cancel 5\cdot \cancel 5\cdot \cancel 5\cdot 5\cdot 5\cdot 5\cdot 5\cdot 5\cdot 5\cdot 5}^\text{10 times}}{\underbrace{\cancel 5\cdot \cancel 5\cdot \cancel 5}_\text{3 times}} \\\\\\ &=\underbrace{5\cdot 5\cdot 5\cdot 5\cdot 5\cdot 5\cdot 5}_\text{7 times} \\\\ &=5^7 \end{aligned}$ In conclusion, $\dfrac{5^{10}}{5^3}=5^7$.